Date of Award

Spring 5-22-2021

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Teaching and Leadership

Advisor(s)

Nicole Fonger

Keywords

Functional Thinking, Networking Theories, Quadratic Functions, Quantitative Reasoning, Representational Fluency, Representations

Subject Categories

Education | Science and Mathematics Education

Abstract

Abstract In this dissertation, I explore ways to support secondary school students’ meaningful understanding of quadratic functions. Specifically, I investigate how students co-developed representational fluency (RF) and functional thinking (FT), when they gained meaningful understanding of quadratic functions. I also characterize students’ co-emergence of RF and FT on each representation (e.g., a graph, a symbolic equation, and a table) and across multiple representations. To accomplish these goals, I employed a design research methodology: a teaching experiment with eight Turkish-American secondary school students in an after-school context at a Turkish Community Center. I constructed the design principles and design elements for the study by networking two distinct domains of literature—representations and quantitative reasoning—to support students’ meaningful learning. I conducted ongoing and retrospective analyses on the enhanced transcriptions of small- and whole-group interactions. The analyses revealed a learning-ecology framework that supported secondary school students’ meaningful understanding of quadratic functions. The learning-ecology framework consisted of three components: enacted task characteristics, teacher pedagogical moves, and socio-mathematical norms. Furthermore, the findings showed that students employed two types of reasoning when they created and connected representations of quantities and the relationships between them: static thinking and lateral thinking. Static thinking is recalling a learned fact to represent a quantitative relationship with no attention to how quantities covary on a representation, while lateral thinking is a creative way of thinking wherein students conceive of concrete representations of functions as an emergent quantitative relationship. The findings also showed that students’ co-emergence of RF and FT can be operationalized into four levels starting from lesser sophisticated reasoning to greater sophisticated reasoning. Level 0 is a disconnection, level 1 is a partial connection, level 2 is a connection and level 3 is flexible a connection between students’ RF and FT. The dissertation informs teachers and the mathematics education community by (a) reporting and verifying the learning-ecology framework that supported students’ meaningful understanding of quadratic functions; and (b) characterizing students’ co-emergence of RF and FT within and across multiple representations.

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